AUTHOR=Sun Wensheng , Yang Yujun 

TITLE=A Note on Resistance Distances of Graphs

JOURNAL=Frontiers in Physics

VOLUME=10

YEAR=2022

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2022.896886

DOI=10.3389/fphy.2022.896886

ISSN=2296-424X

ABSTRACT=<p>Let <italic>G</italic> be a connected graph with vertex set <italic>V</italic>(<italic>G</italic>). The resistance distance between any two vertices <italic>u</italic>, <italic>v</italic> ∈ <italic>V</italic>(<italic>G</italic>) is the net effective resistance between them in the electric network constructed from <italic>G</italic> by replacing each edge with a unit resistor. Let <italic>S</italic> ⊂ <italic>V</italic>(<italic>G</italic>) be a set of vertices such that all the vertices in <italic>S</italic> have the same neighborhood in <italic>G</italic> − <italic>S</italic>, and let <italic>G</italic>[<italic>S</italic>] be the subgraph induced by <italic>S</italic>. In this note, by the {1}-inverse of the Laplacian matrix of <italic>G</italic>, formula for resistance distances between vertices in <italic>S</italic> is obtained. It turns out that resistance distances between vertices in <italic>S</italic> could be given in terms of elements in the inverse matrix of an auxiliary matrix of the Laplacian matrix of <italic>G</italic>[<italic>S</italic>], which derives the reduction principle obtained in [J. Phys. A: Math. Theor. 41 (2008) 445203] by algebraic method.</p>